nLab algebraic Kan complex

Idea

The notion of algebraic Kan complex is an algebraic definition of ∞-groupoids.

It builds on the classical geometric definition of $\infty$-groupoids in terms of Kan complexes. A Kan complex is like an algebraic $\infty$-groupoid in which we have forgotten what precisely the composition operation and what the inverses are, and only know that they do exist. This becomes an algebraic model for $\infty$-groupoids by adding the specific choices of composites back in.

The nontrivial aspect of the definition of algebraic Kan complexes is that they do still present the full (∞,1)-category ∞Grpd. Notably the homotopy hypothesis is true for algebraic Kan complexes.

Definition

An algebraic Kan complex is a Kan complex equipped with a choice of horn fillers for all horns.

A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers.

This defines the category $Alg Kan$ of algebraic Kan complexes.

For more see the section Algebraic fibrant models for higher categories at model structure on algebraic fibrant objects.

A slight variant of this definition is that of a simplicial T-complex.

Properties

Monadicity

The category $Alg Kan$ is the category of algebras over a monad

$$sSet \stackrel{\leftarrow}{\to} Alg sSet \,.$$

This means that algebraic Kan complexes are formally an algebraic model for higher categories.

See model structure on algebraic fibrant objects for details.

Homotopy hypothesis-theorem

The homotopy hypothesis is true for algebraic Kan complexes:

there is a model category structure on $Alg Kan$ – the model structure on algebraic fibrant objects – and a Quillen equivalence to the standard model structure on simplicial sets.

Moreover, there is a direct Quillen equivalence

$$\Pi_\infty : Top \stackrel{\leftarrow}{\to} AlgKan : |-|_r \,,$$

to the standard model structure on topological spaces, where the left adjoint $|-|_r$ is a quotient of the geometric realization of the underlying Kan complexes and $\Pi_\infty$ is a version of the fundamental ∞-groupoid-functor with values in algebraic Kan complexes.

See homotopy hypothesis – for algebraic Kan complexes for details.

Algebraicization

If we assume the axiom of choice, then any Kan complex can be made into an algebraic Kan complex by making a simultaneous choice of a filler for every horn.

In the absence of AC, one might argue that algebraic Kan complexes are a better model of $\infty$-groupoids than non-algebraic ones. For instance, an algebraic Kan complex always has the right lifting property with respect to all anodyne morphisms, whereas for a non-algebraic Kan complex this fact requires choice.

References

• Thomas Nikolaus: Algebraic models for higher categories, Indagationes Mathematicae

  21 1–2 (2011) 52-75 [arXiv:1003.1342, doi:10.1016/j.indag.2010.12.004]